Suppose for the purposes of this question that, for the $5TL$ coins newly put into circulation,

there has been a flaw in manufacturing. Suppose it has been announced that $\frac{1}{4}$ of these coins are

biased; they have a probability $p=\frac{1}{4}$ of landing heads. The remaining $\frac{3}{4}$ of the coins are fair;

they have a probability $p=\frac{1}{2}$ of landing heads. While shopping in a mall, a salesperson hands you

a $5$ TL coin for change, and you want to determine if it is fair or biased.

$($a$)$ Before doing any experiments with the coin, what is your decision? Is the coin fair or

biased? What is the probability of error for your decision?

$($b$)$ Having learned Bayesian decision theory, you know that you can make a better decision

by making observations, i$.$e$.,$ by simply tossing the coin several times and looking at the outcome.

You toss the coin $n$ times, and $k$ out of these $n$ times the coin lands heads. Show that the minimum

probability of error decision rule as a function of $k$ and $n$ is:

Decide

$($c$)$ If $n=10$ and $k=3,$ show that the probability of error is approximately equal to $0\mathrm{.}416,$

according to this decision rule.

Hint: The probability of obtaining $k$ heads in $n$ tosses is $([n],[k])p^k(1-p{)}^{n-k},$ where $p$ is the probability

that the coin lands heads in a single toss.